Cross Multiplication Calculator

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Cross Multiplication (D)

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Table of Contents

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What is Cross Multiplication?

This cross multiplication calculator finds the fourth term in a proportion written as a/b = c/D. You supply a, b and c; the result is D. Proportions describe two equal ratios, so they are useful for scale drawings, recipes, unit rates, percentages and similar problems where one quantity changes in the same ratio as another.

Cross multiplication is a compact algebra step, not a rule for every equation containing fractions. It applies when two ratios are set equal and their denominators are nonzero. Keep each value in its intended position: swapping a numerator and denominator changes the ratio and therefore changes the answer.

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How to Use This Calculator

  1. Write the relationship in the same order as the form: a/b = c/D, with D as the missing denominator.
  2. Enter the known value from the upper left as a, the lower left as b and the upper right as c.
  3. Select Calculate. The calculator multiplies b by c, divides that product by a and rounds the result to four decimal places when needed.
  4. Put the result in D's position and check that both ratios have the same value.

For a real problem, attach the units before entering anything. If 4 notebooks cost $10 and 7 notebooks cost D dollars, the matching order is 4/10 = 7/D only if both numerators count notebooks and both denominators are dollars. The result is then the cost paired with seven notebooks. Consistent placement is more important than which measurement is placed on top.

The calculator accepts decimal and negative values. It does not accept a blank field. Do not enter zero for a because the calculation divides by a. A zero value for b or c is mathematically allowed when a is nonzero and produces D = 0.

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Formula Used

The page uses the proportion a/b = c/D. Cross multiplication gives aD = bc, so the missing term is:

D = (b × c) / a

This rearrangement requires a ≠ 0. The original ratios also require b ≠ 0 and D ≠ 0 if they are to be interpreted as ordinary fractions. The page calculates the rearranged expression directly, so it can display zero when b or c is zero even though substituting D = 0 into c/D may make the original written ratio undefined. For a conventional proportion, use nonzero denominators and verify the completed equality.

A quick check is to compare the cross products after calculation. Multiply a by D and b by c. Allowing for display rounding, those two products should agree. You can also divide a by b and c by D; the two decimal ratios should match.

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Variables Explained

SymbolPosition and meaning
aNumerator of the first ratio and the divisor in this page's rearranged formula. It must not be zero.
bDenominator of the first ratio. It is multiplied by c to find the numerator of the calculation.
cNumerator of the second ratio. Keep its unit aligned with a when modeling a rate.
DThe missing denominator of the second ratio and the displayed result.
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Worked Example

Suppose 4 identical notebooks cost $10 and you want the cost of 7 notebooks. Use notebooks over dollars on both sides:

4/10 = 7/D

Enter a = 4, b = 10 and c = 7. The calculation is D = (10 × 7) / 4 = 70/4 = 17.5. The result means seven notebooks cost $17.50 at the same price per notebook.

Check it two ways. The cross products are 4 × 17.5 = 70 and 10 × 7 = 70. The ratios are also equal: 4/10 = 0.4 and 7/17.5 = 0.4. If you instead wanted the number of notebooks for a known price, rearrange the proportion first so the unknown occupies D's position, then enter the three known terms.

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Why Use This Calculator?

The displayed number belongs in the lower-right position of a/b = c/D. Its unit is normally the same as b because b and D are corresponding terms. A result of 17.5 is not meaningful by itself; in the notebook example it means $17.50 because dollars were placed in both denominators.

Cross multiplication assumes a proportional relationship. It is suitable for constant prices, fixed map scales and recipes scaled without changing ingredient ratios. It is not suitable when there is a fixed starting fee, a changing tax rate, volume discounts, nonlinear growth or any other condition that prevents the ratios from staying equal.

The page rounds to four decimal places. If the result feeds another calculation, reproduce D = bc/a with the full input values rather than copying a rounded display. A repeating result such as 1/3 will appear as 0.3333, so the completed ratios may differ slightly if you check them using only the displayed value.

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Common Mistakes

    • Using the wrong layout: this form solves a/b = c/D. It does not solve a/b = D/c unless you first rearrange your proportion.
    • Mixing corresponding quantities: keep like units in matching numerator or denominator positions.
    • Entering a = 0: the page divides by a, and division by zero has no finite result.
    • Applying cross multiplication to a sum of fractions: first isolate a proportion containing exactly one ratio on each side.
    • Dropping units: a bare answer can hide that terms were placed inconsistently.
    • Rounding early: retain the entered precision until the final answer, especially when the proportion is one step in a longer calculation.
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Frequently Asked Questions

Which missing value does this calculator find?

It finds D in a/b = c/D. Enter the three known terms in their displayed positions. If your unknown is in another position, rearrange the proportion so it becomes the lower-right term before using the form.

How can I check the result without repeating the same division?

Multiply diagonally. The product a × D should equal b × c, apart from any difference caused by the four-decimal display rounding. This check also helps reveal a value entered in the wrong position.

Can I use decimals in a proportion?

Yes. The form accepts decimal inputs. Keep enough precision in measured values, and attach the correct units to each position before interpreting the result.

Why is cross multiplication not valid for every fraction equation?

The cross-product equality follows from two equal ratios. An equation with an added term, such as a/b = c/D + 2, is not in that form. Use algebra to isolate one ratio on each side first.

Can the answer be negative?

Algebraically, yes, when the signs of a, b and c produce a negative quotient. Decide whether a negative value makes sense in the original setting; lengths, item counts and ordinary prices generally cannot be negative.

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Sources & References

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